New solutions of the star-triangle relation with discrete and continuous spin variables

Abstract

A new solution to the star-triangle relation is given, for an Ising type model that involves interacting spins, that contain integer and real valued components. Boltzmann weights of the model are given in terms of the lens elliptic-gamma function, and are based on Yamazaki's recently obtained solution of the star-star relation. The star-triangle given here, implies Seiberg duality for the $4-d$ $\mathcal{N}=1$ $S_1\times S_3/{\mathbb{Z}}_r$ index of the $SU(2)$ quiver gauge theory, and the corresponding two component spin case of the star-star relation of Yamazaki. A proof of the star-triangle relation is given, resulting in a new elliptic hypergeometric integral identity. The star-triangle relation in this paper contains the master solution of Bazhanov and Sergeev as a special case. Two other limiting cases are considered one of which gives a new star-triangle relation in terms of ratios of infinite $q$-products, while the other case gives a new way of deriving a star-triangle relation previously obtained by the author.

18 pages, 1 figure, v2,v3: comments and references added, typos fixed

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Cited by in corpus (21)

• Toroidal and Elliptic Quiver BPS Algebras and Beyond
• The star-triangle relation, lens partition function, and hypergeometric sum/integrals
• The star-triangle relation and 3d superconformal indices
• Elliptic hypergeometric sum/integral transformations and supersymmetric lens index
• Basic hypergeometry of supersymmetric dualities
• On Complex Gamma-Function Integrals
• Barnes-Ismagilov integrals and hypergeometric functions of the complex field
• Integrable quad equations derived from the quantum Yang-Baxter equation
• Integrable lattice spin models from supersymmetric dualities
• Introduction to the theory of elliptic hypergeometric integrals
• Lens partition function, pentagon identity and star-triangle relation
• A New Integrable Ising-type Model from 2d $\mathcal{N}$=(2,2) Dualities
• An analog of the Dougall formula and of the de Branges--Wilson integral
• Hyperbolic and trigonometric hypergeometric solutions to the star-star equation
• Exactly solved models on planar graphs with vertices in ${\mathbb{Z}}^3$
• Two-component Yang-Baxter maps and star-triangle relations
• Lens Generalisation of $\tau$-functions for the Elliptic Discrete Painlevé Equation
• On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$
• Integrability As Duality: The Gauge/YBE Correspondence
• Quasi-classical expansion of a hyperbolic solution to the star-star relation and multicomponent 5-point difference equations